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Properties

Label 3645.2.a.f
Level $3645$
Weight $2$
Character orbit 3645.a
Self dual yes
Analytic conductor $29.105$
Analytic rank $0$
Dimension $9$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3645,2,Mod(1,3645)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3645, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3645.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3645 = 3^{6} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3645.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(29.1054715368\)
Analytic rank: \(0\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - 3x^{8} - 9x^{7} + 28x^{6} + 18x^{5} - 63x^{4} - 2x^{3} + 30x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{7} + \beta_{5} + \beta_{4} + \beta_1 + 1) q^{4} - q^{5} + ( - \beta_{8} + \beta_{7} - \beta_{6}) q^{7} + (\beta_{7} - \beta_{6} + \beta_{4} + \beta_{2} + 3 \beta_1 + 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{7} + \beta_{5} + \beta_{4} + \beta_1 + 1) q^{4} - q^{5} + ( - \beta_{8} + \beta_{7} - \beta_{6}) q^{7} + (\beta_{7} - \beta_{6} + \beta_{4} + \beta_{2} + 3 \beta_1 + 1) q^{8} - \beta_1 q^{10} + ( - \beta_{8} + \beta_{6} + \beta_{5} - \beta_1 + 2) q^{11} + ( - \beta_{8} - \beta_{7} + \beta_{3} + \beta_{2} + \beta_1 - 1) q^{13} + (2 \beta_{7} - \beta_{6} + \beta_{5} - 2 \beta_{2} - 2 \beta_1 + 1) q^{14} + (\beta_{8} + \beta_{7} - \beta_{6} + 3 \beta_{5} + \beta_{4} - 2 \beta_{3} + 3 \beta_1 + 3) q^{16} + ( - \beta_{8} + 2 \beta_{7} - \beta_{6} + \beta_{4} + \beta_{3} + 3) q^{17} + (2 \beta_{7} + \beta_{6} + \beta_{5} - \beta_1) q^{19} + ( - \beta_{7} - \beta_{5} - \beta_{4} - \beta_1 - 1) q^{20} + ( - 2 \beta_{7} - 2 \beta_{5} - \beta_{4} + \beta_{3} + 2 \beta_1 - 2) q^{22} + ( - \beta_{8} + 2 \beta_{7} - \beta_{6} + 2 \beta_{5} + \beta_{4} + \beta_{2} + \beta_1 + 3) q^{23} + q^{25} + (\beta_{8} - 3 \beta_{7} + \beta_{6} + 3 \beta_{5} + \beta_{4} - \beta_1) q^{26} + (\beta_{7} - 5 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} - \beta_{2} - \beta_1 - 2) q^{28} + ( - \beta_{8} + \beta_{7} - \beta_{6} + \beta_{4} - \beta_{3} + 2 \beta_{2}) q^{29} + ( - \beta_{5} + 2 \beta_{4} + \beta_{3} + 3 \beta_1 - 2) q^{31} + (\beta_{7} + \beta_{6} + 3 \beta_{5} + \beta_{4} - \beta_{3} + 2 \beta_{2} + 5 \beta_1 + 3) q^{32} + (2 \beta_{7} - 2 \beta_{6} - 2 \beta_{2} + 3 \beta_1 - 1) q^{34} + (\beta_{8} - \beta_{7} + \beta_{6}) q^{35} + (\beta_{4} - \beta_{3} + \beta_1 + 4) q^{37} + (2 \beta_{7} - 2 \beta_{6} - 2 \beta_{5} - \beta_{4} - 2 \beta_{3} - \beta_{2} + \beta_1 - 1) q^{38} + ( - \beta_{7} + \beta_{6} - \beta_{4} - \beta_{2} - 3 \beta_1 - 1) q^{40} + ( - \beta_{8} + \beta_{7} + \beta_{6} + \beta_{5} + \beta_{3} - \beta_{2} + 1) q^{41} + ( - 2 \beta_{8} + \beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} + \beta_{2} + \beta_1 + 1) q^{43} + (2 \beta_{8} + \beta_{5} + 2 \beta_{4} + \beta_{3} - \beta_{2} - 2 \beta_1 + 3) q^{44} + (\beta_{8} + \beta_{7} - 2 \beta_{6} + 3 \beta_{5} + \beta_{4} - 2 \beta_{3} + 6 \beta_1 - 1) q^{46} + ( - \beta_{8} - 4 \beta_{7} + \beta_{6} - 2 \beta_{5} - 2 \beta_{3} + 2 \beta_{2} + \beta_1 - 1) q^{47} + (2 \beta_{8} - 3 \beta_{7} + 3 \beta_{6} + \beta_{5} - \beta_{4} - 3 \beta_{3} + \beta_{2} - \beta_1 - 1) q^{49} + \beta_1 q^{50} + (2 \beta_{8} - 9 \beta_{7} + 3 \beta_{6} - 3 \beta_{5} - \beta_{4} + \beta_{3} + 6 \beta_{2} + \cdots - 9) q^{52}+ \cdots + (\beta_{8} - 5 \beta_{7} + 3 \beta_{6} - \beta_{5} - \beta_{4} - 2 \beta_{3} + 5 \beta_{2} + 2 \beta_1 - 3) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + 3 q^{2} + 9 q^{4} - 9 q^{5} - 9 q^{7} + 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 9 q + 3 q^{2} + 9 q^{4} - 9 q^{5} - 9 q^{7} + 12 q^{8} - 3 q^{10} + 15 q^{11} - 6 q^{13} - 6 q^{14} + 33 q^{16} + 15 q^{17} - 6 q^{19} - 9 q^{20} - 6 q^{22} + 18 q^{23} + 9 q^{25} + 12 q^{26} - 24 q^{28} - 9 q^{29} - 9 q^{31} + 42 q^{32} - 12 q^{34} + 9 q^{35} + 39 q^{37} - 18 q^{38} - 12 q^{40} + 6 q^{41} + 6 q^{43} + 27 q^{44} + 3 q^{46} + 6 q^{47} + 12 q^{49} + 3 q^{50} - 27 q^{52} + 3 q^{53} - 15 q^{55} + 12 q^{56} - 33 q^{58} + 3 q^{59} - 18 q^{61} + 54 q^{62} + 54 q^{64} + 6 q^{65} + 3 q^{67} + 48 q^{68} + 6 q^{70} + 18 q^{71} - 42 q^{73} + 36 q^{74} + 51 q^{76} - 15 q^{77} - 3 q^{79} - 33 q^{80} + 24 q^{82} + 15 q^{83} - 15 q^{85} + 36 q^{86} - 60 q^{88} + 15 q^{89} + 33 q^{91} + 87 q^{92} + 36 q^{94} + 6 q^{95} + 9 q^{97} + 6 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - 3x^{8} - 9x^{7} + 28x^{6} + 18x^{5} - 63x^{4} - 2x^{3} + 30x^{2} - 3 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -9\nu^{8} + 30\nu^{7} + 71\nu^{6} - 246\nu^{5} - 80\nu^{4} + 386\nu^{3} - 170\nu^{2} - 154\nu + 81 ) / 89 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -7\nu^{8} + 53\nu^{7} - 14\nu^{6} - 488\nu^{5} + 541\nu^{4} + 1032\nu^{3} - 1220\nu^{2} - 278\nu + 241 ) / 89 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 12\nu^{8} - 40\nu^{7} - 65\nu^{6} + 328\nu^{5} - 190\nu^{4} - 485\nu^{3} + 909\nu^{2} - 210\nu - 375 ) / 89 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 20\nu^{8} - 37\nu^{7} - 227\nu^{6} + 339\nu^{5} + 781\nu^{4} - 749\nu^{3} - 888\nu^{2} + 362\nu + 176 ) / 89 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -29\nu^{8} + 67\nu^{7} + 298\nu^{6} - 585\nu^{5} - 861\nu^{4} + 1046\nu^{3} + 807\nu^{2} + 18\nu - 273 ) / 89 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -32\nu^{8} + 77\nu^{7} + 292\nu^{6} - 667\nu^{5} - 591\nu^{4} + 1234\nu^{3} + 68\nu^{2} - 241\nu - 68 ) / 89 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -83\nu^{8} + 247\nu^{7} + 724\nu^{6} - 2239\nu^{5} - 1252\nu^{4} + 4608\nu^{3} - 480\nu^{2} - 1440\nu + 213 ) / 89 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} + \beta_{5} + \beta_{4} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} - \beta_{6} + \beta_{4} + \beta_{2} + 7\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{8} + 7\beta_{7} - \beta_{6} + 9\beta_{5} + 7\beta_{4} - 2\beta_{3} + 9\beta _1 + 17 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 9\beta_{7} - 7\beta_{6} + 3\beta_{5} + 9\beta_{4} - \beta_{3} + 10\beta_{2} + 49\beta _1 + 11 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 10\beta_{8} + 46\beta_{7} - 9\beta_{6} + 67\beta_{5} + 49\beta_{4} - 20\beta_{3} + 3\beta_{2} + 74\beta _1 + 109 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 3\beta_{8} + 67\beta_{7} - 46\beta_{6} + 40\beta_{5} + 74\beta_{4} - 13\beta_{3} + 80\beta_{2} + 349\beta _1 + 98 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 80 \beta_{8} + 302 \beta_{7} - 67 \beta_{6} + 481 \beta_{5} + 349 \beta_{4} - 156 \beta_{3} + 50 \beta_{2} + 592 \beta _1 + 730 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−2.48177
−1.62007
−0.612073
−0.371834
0.368923
0.825747
1.45981
2.66571
2.76555
−2.48177 0 4.15918 −1.00000 0 −3.56739 −5.35860 0 2.48177
1.2 −1.62007 0 0.624620 −1.00000 0 1.63015 2.22821 0 1.62007
1.3 −0.612073 0 −1.62537 −1.00000 0 2.07603 2.21899 0 0.612073
1.4 −0.371834 0 −1.86174 −1.00000 0 −2.73728 1.43593 0 0.371834
1.5 0.368923 0 −1.86390 −1.00000 0 2.37541 −1.42548 0 −0.368923
1.6 0.825747 0 −1.31814 −1.00000 0 −4.92678 −2.73995 0 −0.825747
1.7 1.45981 0 0.131053 −1.00000 0 −1.54557 −2.72831 0 −1.45981
1.8 2.66571 0 5.10601 −1.00000 0 1.38284 8.27973 0 −2.66571
1.9 2.76555 0 5.64827 −1.00000 0 −3.68741 10.0895 0 −2.76555
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3645.2.a.f yes 9
3.b odd 2 1 3645.2.a.e ✓ 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3645.2.a.e ✓ 9 3.b odd 2 1
3645.2.a.f yes 9 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{9} - 3T_{2}^{8} - 9T_{2}^{7} + 28T_{2}^{6} + 18T_{2}^{5} - 63T_{2}^{4} - 2T_{2}^{3} + 30T_{2}^{2} - 3 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3645))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{9} - 3 T^{8} - 9 T^{7} + 28 T^{6} + \cdots - 3 \) Copy content Toggle raw display
$3$ \( T^{9} \) Copy content Toggle raw display
$5$ \( (T + 1)^{9} \) Copy content Toggle raw display
$7$ \( T^{9} + 9 T^{8} + 3 T^{7} - 146 T^{6} + \cdots + 3048 \) Copy content Toggle raw display
$11$ \( T^{9} - 15 T^{8} + 45 T^{7} + \cdots + 4401 \) Copy content Toggle raw display
$13$ \( T^{9} + 6 T^{8} - 69 T^{7} + \cdots - 14867 \) Copy content Toggle raw display
$17$ \( T^{9} - 15 T^{8} + 42 T^{7} + \cdots + 159 \) Copy content Toggle raw display
$19$ \( T^{9} + 6 T^{8} - 99 T^{7} + \cdots + 628021 \) Copy content Toggle raw display
$23$ \( T^{9} - 18 T^{8} + 39 T^{7} + \cdots + 292599 \) Copy content Toggle raw display
$29$ \( T^{9} + 9 T^{8} - 111 T^{7} + \cdots - 68799 \) Copy content Toggle raw display
$31$ \( T^{9} + 9 T^{8} - 117 T^{7} + \cdots - 546984 \) Copy content Toggle raw display
$37$ \( T^{9} - 39 T^{8} + 636 T^{7} + \cdots - 719 \) Copy content Toggle raw display
$41$ \( T^{9} - 6 T^{8} - 90 T^{7} + \cdots - 7299 \) Copy content Toggle raw display
$43$ \( T^{9} - 6 T^{8} - 141 T^{7} + \cdots + 139231 \) Copy content Toggle raw display
$47$ \( T^{9} - 6 T^{8} - 276 T^{7} + \cdots - 2286897 \) Copy content Toggle raw display
$53$ \( T^{9} - 3 T^{8} - 264 T^{7} + \cdots + 67017 \) Copy content Toggle raw display
$59$ \( T^{9} - 3 T^{8} - 141 T^{7} + \cdots + 963 \) Copy content Toggle raw display
$61$ \( T^{9} + 18 T^{8} - 123 T^{7} + \cdots + 30949993 \) Copy content Toggle raw display
$67$ \( T^{9} - 3 T^{8} - 345 T^{7} + \cdots + 5755641 \) Copy content Toggle raw display
$71$ \( T^{9} - 18 T^{8} - 96 T^{7} + \cdots + 24570867 \) Copy content Toggle raw display
$73$ \( T^{9} + 42 T^{8} + 354 T^{7} + \cdots - 41719697 \) Copy content Toggle raw display
$79$ \( T^{9} + 3 T^{8} - 429 T^{7} + \cdots + 18385701 \) Copy content Toggle raw display
$83$ \( T^{9} - 15 T^{8} - 207 T^{7} + \cdots - 15514497 \) Copy content Toggle raw display
$89$ \( T^{9} - 15 T^{8} - 369 T^{7} + \cdots - 24482709 \) Copy content Toggle raw display
$97$ \( T^{9} - 9 T^{8} - 645 T^{7} + \cdots + 67355713 \) Copy content Toggle raw display
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